Theorem hbxfreq 2185

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1401 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Hypotheses

Ref	Expression
hbxfr.1	|- A = B
hbxfr.2	|- ( y e. B -> A. x y e. B )

Assertion

Ref	Expression
hbxfreq	|- ( y e. A -> A. x y e. A )

Proof of Theorem hbxfreq

Step	Hyp	Ref	Expression
1		hbxfr.1	. . 3 |- A = B
2	1	eleq2i 2145	. 2 |- ( y e. A <-> y e. B )
3		hbxfr.2	. 2 |- ( y e. B -> A. x y e. B )
4	2, 3	hbxfrbi 1401	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by: (None)